whenever \(0\le \mu (m) < r < t,\)  and \(s \ge 0.\)  The cases where \(t\le 4\)  are known by Peterson [1]}, Kameko [2]} and Sum [3]}. For \(t = 5\) , it has been partially studied by us [4]}, [5]}, [6]}, [7]}, [8]} and others. The problem is not yet solved in general. The present Note is motivated by the previous results and we here continue this line of investigation by considering \((QP_t)_n\)  in degrees \(n\)  of the form (REF ). More precisely, when \(t = 5,\)  we explicitly determine the structure of \((QP_5)_n\)  for \(n_s :=n= 5(2^{s}-1) + 42.2^{s}\). As a result, it confirms Sum's conjecture [8]} for the relationship between the minimal sets of \(\mathcal {A}_2\) -generators of the polynomial rings \(P_{t-1}\)  and \(P_{t}\)  in the case \(t=5\)  and degree \(n_s.\)  An efficient approach to solve the hit problem of five variables had been provided.
As applications, we obtain the dimension of \((QP_6)_n\)  in degree \(5(2^{s+5}-1) + n_0.2^{s+5}\) for all \(s\ge 0\) , and show that Singer's conjecture [10]} for the fifth cohomological transfer holds in degrees \(n_s.\)
